Li Yi

Li Yi

I am a 3rd year PhD student majoring in birational geometry

Metric method in Birational Geometry

less than 1 minute read

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In this series of notes we will summarize some standard metric methods which will be useful in birational geometry.

Ohsawa-Takegoshi extension theorem with variants

1. The Ohsawa–Takegoshi extension theorem and why it’s uesful in birational geometry [upd 10.8]

Metric Methods in Invariance of Plurigenera

1. Siu’s invariance of plurigenera analytic proof

2. Limiting metric arguement in birational geometry

Analytic Methods for Positivities of Direct image and Canonical Bundle Formulas

1. Paun-Takayama’s construction of singular Hermitian metric on direct image of relative canonical sheaf

2. Hacon-Popa-Schnell’s construction of singular Hermitian metric on direct image of relative (pluri)canonical sheaves [upd 10.12]

3. Hacon-Paun’s metric method for canonical bundle formula

Applications of positivities of direct images

1. Kollar’s approach to Projectivity of moduli

2. Positivity of direct image in Iitaka conjecture

Covering Tricks of Kawamata, Viehweg

Metric Methods in Abundance Conjecture

1. Supercanonical metric and abundance conjecture

2. Siu and Paun’s analytic proof of finite generation of canonical ring and Shokurov’s non-vanishing

Metric Methods in Boundedness Problems

1. The theorem of Angehrn and Siu

2. Birational boundedness for varieties of general type

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The aim of this note is to introduce the minimal model program for Kahler varieties. Compared with the well-known minimal model program for projective varieties, the theory of Kahler minimal model program will have the following three major difficulties: (1) Mori bend-and-break technique: Mori bend and break is not known for Kahler varieties, (2) The base point free theorem and cone theorem: Since Kahler manifold with a big line bundle is automatic projective, thus a big line bundle does not make sense on (non-projective) Kahler manifold, (3) Contraction theorem: In the classical MMP, we require the base point free theorem to produce some semi-ampleness divisor. This semi-ample divisor will induce the negative contraction morphism, and this approach is not very clear under Kaehler setting.

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This aim of this series of notes is to study the classical paper BCHM in details. We will try to finish the proof of the main theorem and summarize some interesting applications of BCHM. We will not strictly follow the structure of the original paper, instead we will divide the paper into several topics discussion, including: Existence of flip, existence of minimal model, termination problem, non-vanishing conjecture and finite generation problem etc.

Birational Geometry Notes: Around Moduli theory

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In this part of the notes, we will focus on the properties of moduli spaces. The guiding philosophy behind moduli theory is that

The moduli of a geometric object completely determine the object itself. In birational geometry, one of the central theme is that the geometry of a variety is largely governed by the curves and divisors it contains.